Solution
NI's low-cost DAQ such as USB-6001, USB-6002 or USB-6003 can only outputting limited amount of current which might not be enough to drive the relay. You need to check the device specification and make sure the current amount is sufficient to drive the relay.
For example, the below image is taken from USB-6001 specification sheet that show digital output maximum current per line is limited to 4mA while OMRON G3S4‑D1 required at least 12.5mA current.
Reference: G3S4 Terminal SSR/Specifications | OMRON Industrial Automation
In order to control relay with USB-6001, you can use custom external circuit to amplify the current such as Darlington transistor circuit.
Alternatively, you can use other DAQ board that can output higher current that match your relay requirement.